How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$?
We recognize 8, 15, and 17 as a Pythagorean triple. Since the hypotenuse is the longest side of the right triangle, the altitude to the hypotenuse is the shortest of the altitudes. The other two altitudes are just the legs themselves, therefore $8 + 15 = \boxed{23}.$